Shevchenko S. Geometric modeling of Resonance of a Swing Spring

The object of the research is the process of constructing the trajectory of the load of a swinging spring with a movable suspension point in order to identify and analyze its resonance. The aim of the study is to develop a method for geometric modeling of the resonance of a swinging spring based on the construction of the trajectory of its load taking into account the initial conditions for the occurrence of oscillations. Methods of mathematical modeling, analytical and computational geometry, geometric modeling and design, differential equations were used. A method has been developed for calculating the values of parameters and initial conditions under which the trajectory of movement of the swinging spring load (including those with a movable suspension point) will have the form of a periodic one. A table of geometric shapes of periodic trajectories of movement of the load of swinging springs, corresponding to the selected values of stiffness and mass, has been compiled. Phase trajectories of functions of generalized coordinates of a swinging spring and its varieties are constructed and estimates of the range of changes in their values and velocities of the load are provided. A set of trajectories of movement of the swinging spring load have been determined, which will correspond to a given ratio of vertical and horizontal frequencies of the load oscillations. The method for determining the numbers that characterize the obtained geometric forms of periodic trajectories of movement and corresponding to a given ratio of horizontal and vertical periods of cargo oscillations has been improved. The solution of a class of problems related to the process of movement of a swinging spring, taking into account the flow of horizontal vibrations into vertical ones and vice versa, was further developed. The solutions of problems on the subject of emergency rescue services are given. The research results were transferred for use at a number of enterprises and in the educational process of the National University of Civil Defenсe of Ukraine.